Unlike systems of linear equations, systems of multivariate polynomial
equations over the complex numbers or finite fields can be compactly used to
model combinatorial problems. In this way, a problem is feasible (e.g. a
graph is 3-colorable, Hamiltonian, etc.) if and only if a given system of
polynomial equations has a solution. Via Hilbert’s Nullstellensatz, we
generate a sequence of large-scale, sparse linear algebra computations from
these non-linear models to describe an algorithm for solving the underlying
combinatorial problem. As a byproduct of this algorithm, we produce algebraic
certificates of the non-existence of a solution (i.e., non-3-colorability,
non-Hamiltonicity, or non-existence of an independent set of size k).

In this talk, we present theoretical and experimental results on the size of
these sequences, and the complexity of the Hilbert’s Nullstellensatz
algebraic certificates. For non-3-colorability over a finite field, we
utilize this method to successfully solve graph problem instances having
thousands of nodes and tens of thousands of edges. We also describe methods
of optimizing this method, such as finding alternative forms of the
Nullstellensatz, adding carefully-constructed polynomials to the system,
branching and exploiting symmetry.